Whakaoti i te 5x^2+60x+180=0

$$5 x ^ { 2 } + 60 x + 180 = 0$$

Whakaoti mō x

$$x=-6$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Whakawehea ngā taha e rua ki te 5.

$$x^{2}+12x+36=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei x^{2}+ax+bx+36. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=12 ab=1\times 36=36$$

I te mea kua tōrunga te ab, he ōrite te tohu o a me b. I te mea kua tōrunga te a+b, he tōrunga hoki a a me b. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua 36.

$$1,36 2,18 3,12 4,9 6,6$$

Tātaihia te tapeke mō ia takirua.

$$1+36=37 2+18=20 3+12=15 4+9=13 6+6=12$$

Ko te otinga te takirua ka hoatu i te tapeke 12.

$$a=6 b=6$$

Tuhia anō te x^{2}+12x+36 hei \left(x^{2}+6x\right)+\left(6x+36\right).

$$\left(x^{2}+6x\right)+\left(6x+36\right)$$

Tauwehea te x i te tuatahi me te 6 i te rōpū tuarua.

$$x\left(x+6\right)+6\left(x+6\right)$$

Whakatauwehea atu te kīanga pātahi x+6 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x+6\right)\left(x+6\right)$$

Tuhia anōtia hei pūrua huarua.

$$\left(x+6\right)^{2}$$

Hei kimi i te otinga whārite, whakaotia te x+6=0.

$$x=-6$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Ko ngā whārite katoa o te āhua ax^{2}+bx+c=0 ka taea te whakaoti mā te whakamahi i te tikanga tātai pūrua: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. E rua ngā otinga ka puta i te tikanga tātai pūrua, ko tētahi ina he tāpiri a ±, ā, ko tētahi ina he tango.

$$5x^{2}+60x+180=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 5 mō a, 60 mō b, me 180 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-60±\sqrt{60^{2}-4\times 5\times 180}}{2\times 5}$$

Pūrua 60.

$$x=\frac{-60±\sqrt{3600-4\times 5\times 180}}{2\times 5}$$

Whakareatia -4 ki te 5.

$$x=\frac{-60±\sqrt{3600-20\times 180}}{2\times 5}$$

Whakareatia -20 ki te 180.

$$x=\frac{-60±\sqrt{3600-3600}}{2\times 5}$$

Tāpiri 3600 ki te -3600.

$$x=\frac{-60±\sqrt{0}}{2\times 5}$$

Tuhia te pūtakerua o te 0.

$$x=-\frac{60}{2\times 5}$$

Whakareatia 2 ki te 5.

$$x=-\frac{60}{10}$$

Whakawehe -60 ki te 10.

$$x=-6$$

Ngā Upane mō te Whakaoti i te Pūrua

Ko ngā whārite pūrua pēnei i tēnei nā ka taea te whakaoti mā te whakaoti i te pūrua. Hei whakaoti i te pūrua, ko te whārite me mātua tuhi ki te āhua x^{2}+bx=c.

$$5x^{2}+60x+180=0$$

Me tango 180 mai i ngā taha e rua o te whārite.

$$5x^{2}+60x+180-180=-180$$

Mā te tango i te 180 i a ia ake anō ka toe ko te 0.

$$5x^{2}+60x=-180$$

Whakawehea ngā taha e rua ki te 5.

$$\frac{5x^{2}+60x}{5}=-\frac{180}{5}$$

Mā te whakawehe ki te 5 ka wetekia te whakareanga ki te 5.

$$x^{2}+\frac{60}{5}x=-\frac{180}{5}$$

Whakawehe 60 ki te 5.

$$x^{2}+12x=-\frac{180}{5}$$

Whakawehe -180 ki te 5.

$$x^{2}+12x=-36$$

Whakawehea te 12, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te 6. Nā, tāpiria te pūrua o te 6 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+12x+6^{2}=-36+6^{2}$$

Pūrua 6.

$$x^{2}+12x+36=-36+36$$

Tāpiri -36 ki te 36.

$$x^{2}+12x+36=0$$

Tauwehea x^{2}+12x+36. Ko te tikanga pūnoa, ina ko x^{2}+bx+c he pūrua tika pūrua tika pū, ka taea taua mea te tauwehea i ngā wā katoa hei \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+6\right)^{2}=0$$

Tuhia te pūtakerua o ngā taha e rua o te whārite.

$$\sqrt{\left(x+6\right)^{2}}=\sqrt{0}$$

Whakarūnātia.

$$x+6=0 x+6=0$$

Me tango 6 mai i ngā taha e rua o te whārite.

$$x=-6 x=-6$$

Kua oti te whārite te whakatau. He ōrite ngā whakatau.

$$x=-6$$